# Thread: Multivariable problem involving integrals

1. ## Multivariable problem involving integrals

I'm having problems with this and was looking for some insight.

"Randy is going to use a porcelain bowl to measure rain. The bowl has the shape $\displaystyle z=x^2+y^2$ from $\displaystyle z=0$ to $\displaystyle z=10$ inches. Randy must calibrate the bowl so that he can precisely measure the amount of rain that falls. What height of water in the bowl corresponds to an inch of rain? That is to say, a circular cylinder would have one inch of rain in it, or that the level ground would have standing water one inch deep."

Are there any ideas?

2. Work out the volume $\displaystyle V$ in the bowl up to a height $\displaystyle h$ in cylindrical polars as

$\displaystyle V = \int_0^h \pi r^2 \, \mathrm{d} z$

since the volume of a slice of thickness $\displaystyle dz$ would have volume $\displaystyle \pi r^2 \mathrm{d}z$ where $\displaystyle r^2 = z$ so

$\displaystyle V = \pi \int_0^h z \, \mathrm{d} z = \frac{\pi}{2} h^2$.

The volume, $\displaystyle V_1$ ,representing one inch of rain is the volume in a cylindrical container of the same cross-sectional area as the opening of the bowl. So

$\displaystyle V_1 = \pi R^2 \cdot 1$

where $\displaystyle R$ is the radius at the opening of the bowl and since $\displaystyle z=R^2$ when $\displaystyle z=10$ we have

$\displaystyle V_1 = 10 \pi$.

Equating so $\displaystyle V_1= V$ we have

$\displaystyle 10 \pi = \frac{\pi}{2} h^2$,

so

$\displaystyle h = \sqrt{20} \quad \text{inches}$.